The generator matrix

 1  0  0  1  1  1 X+2 3X  1  1 3X+2  1  0  1  1 X+2  1  X  1 2X  2  1 2X+2  1  1 X+2  0 2X+2  1  1  1  1 3X  X  0  1  1  1  X  1  1  1 2X+2  1  1  1 2X+2 2X  1  1  2  1 X+2  1 X+2 2X+2  2  1 3X  1  1  1  1  1 3X  1 2X  1
 0  1  0  0 2X+3 X+1  1 2X+2 3X 2X+3  1  X  1  3 3X+3  1 3X 3X+2 2X+3 2X+2  1 3X+2  1 3X+3 2X  1 3X  1 3X+2  0 3X+3  3 2X  1  1 X+1 3X 2X+2  1  3 3X+1 2X+2  1  0 3X+3 2X+1  1  1  3 2X  1 X+1 3X+2 3X 3X  1 2X+2 2X  1  X 3X+2 X+3 X+3  X  1 2X  1 3X+2
 0  0  1  1  1  0 2X+3  1 3X 3X 2X 2X+3 3X+2 3X+1 3X+3 3X+1 X+1  1 2X+2  1 2X+3 2X X+1  X X+3  1  1  2 3X 3X 2X+3  3  1  3  X  2 2X+1 3X+1  0  0 X+1 2X+2 3X+1 3X+1  X X+2 X+1 3X+2  2  1 3X 2X  1 X+3  1 2X+3  1 2X+1 2X 3X+2 2X 3X+1  2  2 X+2 3X+3 3X+2  0
 0  0  0  X 3X 2X 3X  X 2X+2  2  0  X 2X+2 3X+2 3X+2 X+2 X+2  X 2X 3X 3X+2 2X+2 3X 2X 3X 3X+2 3X+2 2X+2 X+2 3X+2  0  0 2X+2  0  X 3X+2  2  2 X+2  X  2 X+2  2 2X 3X  0 2X 3X+2  2  0  0  X  2 2X 2X  X  2 X+2  2  0 3X+2 2X 2X+2 3X  2  2  X  2

generates a code of length 68 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 62.

Homogenous weight enumerator: w(x)=1x^0+424x^62+1332x^63+2216x^64+2860x^65+3516x^66+4180x^67+4212x^68+4386x^69+3235x^70+2612x^71+1799x^72+938x^73+527x^74+252x^75+163x^76+70x^77+25x^78+8x^79+8x^80+2x^81+1x^82+1x^84

The gray image is a code over GF(2) with n=544, k=15 and d=248.
This code was found by Heurico 1.16 in 11.4 seconds.